Questions about the branch of abstract algebra that deals with groups.

**4**

votes

**1**answer

187 views

### Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as
\begin{equation*}
d_\lambda(A) := \sum_{\sigma \in S_n} ...

**2**

votes

**1**answer

173 views

### Normal subgroup of a totally ordered group

A totally ordered group is a group equipped with a compatible total order, that is, $x\leq y$ and $z\leq t$ imply $x+z\leq y+t$ for all $x,y,z,t$ in the group.
Is it true that every totally ordered ...

**0**

votes

**1**answer

107 views

### Unipotent orbit in adjoint group over finite field

[Editted: The assertion is wrong; see Jay's answer]
My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...

**3**

votes

**2**answers

271 views

### Adeles and twisted adeles

Let $\mu_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu_n=\ker[{\mathbb{C}}^*\overset{n}{\longrightarrow}{\mathbb{C}}^*]$.
We set
$$ \mu=\varinjlim_n \mu_n\subset ...

**6**

votes

**1**answer

199 views

### Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of ...

**7**

votes

**1**answer

247 views

### Can you decide whether the commutator subgroup of a f.p. group is f.g?

Is the following algorithmic problem known to be decidable/undecidable?
Input: a finite group presentation $P$.
Decide: is the commutator subgroup of the group presented by $P$ finitely generated?

**10**

votes

**1**answer

353 views

### Number of trivializations of a trivial word in the free group

Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...

**4**

votes

**1**answer

206 views

### Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...

**4**

votes

**1**answer

164 views

### Finite p-groups and their fibered products

Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?

**12**

votes

**2**answers

389 views

### generating set for symmetric group $S_n$

Say that $a_1, \cdots, a_{n-1}$ is an independent generating set for $S_n$. Let $b$ be any element in $S_n$. Is it true that $b$ can replace one of the generators, i.e. that there exists an index $i$, ...

**4**

votes

**2**answers

162 views

### Centralizers of reflections in special subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with ...

**4**

votes

**2**answers

279 views

### Maximal abelian subgroup of general linear groups

Thanks for any help or comments.
Is it possible to recognize all maximal abelian subgroups of general linear group on finite field $F$ of order $q$, $GL_n(F)$.
By maximal abelian I mean if $A$ is ...

**3**

votes

**1**answer

177 views

### About the second largest adjacency eigenvalue of Abelian Cayley graphs

[Assume all groups are finite]
One knows the general statement that the sum of the values of the character function on the generating set is an eigenvalue of a Cayley graph.
But the above doesn't ...

**2**

votes

**0**answers

90 views

### project limit on $n$- simplical complex which is principal homogeneous with respect to an action

The setting:
Let G be compact locally $\Bbb{Q}_p$ analytic group. We fix a countable basis of open normal subgroups $G\supset G_1\supset ...G_r\supset...$
We suppose that we are given a system of ...

**3**

votes

**1**answer

135 views

### Subgroups of index 2 in a fibered product

Let $G$ be a finite group and let $M,N \lhd G$ be normal subgroups with a trivial intersection. Suppose that $G$ has a subgroup of index $2$. Must $G$ have a subgroup of index $2$ which contains ...

**2**

votes

**1**answer

113 views

### Is there a non right-orderable torsion-free factor of the Braid group on 3 strands?

The braid group on 3 strands has the presentation $\langle x,y \;|\; xyx=yxy\rangle$. A group $G$ is called right orderable if there is a total order $<$ on the set $G$ such that if $a<b$ then ...

**4**

votes

**1**answer

215 views

### Quotient of principal congruence subgroups

This is a direct follow-up to this question. What is the quotient $\Gamma(2)/\Gamma(2^n)?$ (the principal congruence subgroups are in $SL(2, \mathbb{Z}).$ It is a 2-group, but what else?

**5**

votes

**4**answers

758 views

### Consequences of the Inverse Galois Problem

Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. ...

**3**

votes

**0**answers

98 views

### uniqueness of quotients of principal congruence subgroups

For each $n \geq 2$, is $\Gamma(2^{n})$ the unique normal subgroup of $\Gamma(2)$ with quotient isomorphic to $\Gamma(2) / \Gamma(2^{n})$ (here we are talking about principal congruence subgroups of ...

**0**

votes

**1**answer

202 views

### Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.

**4**

votes

**1**answer

147 views

### Motivational ideas for the Gelfand-Graev character of a finite group of Lie type

I've been studying the Gelfand-Graev character's general construction for a finite group of Lie type. I wish to discuss its particularization in a seminar for the general linear group over a finite ...

**28**

votes

**2**answers

1k views

### How do you *state* the Classification of finite simple groups?

From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...

**4**

votes

**1**answer

171 views

### About the set of Sylow-$p$ subgroups of $G$

Let G be a finite group and S be the set of Sylow p-subgroups of G for a
prime p dividing the order of G. Assume that |S|>1.
Let U and V be two disjoint non-empty subsets of S such that,
...

**1**

vote

**0**answers

64 views

### Counting group invariants using Macdonald conjecture?

It is known from Dyson, Macdonald $et$ $al$ that the constant term asscoiated to the expansion of the following expression
\begin{equation}
\prod_{\alpha \in \Delta} (1-e^{\alpha})^k
\end{equation}
...

**3**

votes

**1**answer

101 views

### Section of Cayley graphs

Let $\pi: G\to G_1$ be a surjective group homomorphism to a finite group $G_1$ and let $S_1$ be a (finite) generating set of $G_1$. Assume $\mathrm{Cay}(G_1,S_1)$ is the Cayley graph of $G_1$ with ...

**3**

votes

**0**answers

175 views

### every element with eigenvalue 1

Is there a "non-trivial" subgroup $G$ of $GL(n,\mathbb C)$ whose every element has an eigenvalue $1$?
"Non-trivial" here means that is no common eigenvector with eigenvalue $1$ for all elements of ...

**14**

votes

**1**answer

730 views

### In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = ...

**7**

votes

**3**answers

370 views

### Decision problem on triviality of intersection of two subgroups

What is known about the following decision problem?
Given two finite sets in a finitely generated group G,
decide whether the subgroups generated by them have trivial intersection.
Is this problem ...

**3**

votes

**1**answer

192 views

### Conjugation of group extensions

Let $H$ be a finite group. We write ${{\mathbb{C}}}^{*n}$ for the $n$-dimensional complex torus $({{\mathbb{C}}}^*)^n$.
We have a short exact sequence
$$ 0\to {{\mathbb{Z}}}^n\to ...

**10**

votes

**2**answers

462 views

### History of Tarski's problems on free groups

As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original ...

**5**

votes

**1**answer

339 views

### When are (Abelian) Cayley graphs also expanders?

I want to ask the question in two parts,
(1)
Is there some fundamental distinguishing property between Abelian and non-Abelian Cayley graphs? (say some specific proof technique which distinguishes ...

**6**

votes

**0**answers

291 views

### Does the Approximation Property (AP) pass to quotients by amenable subgroups?

Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP?
In particular, does there exist a group $G$ with the AP and a surjective group ...

**4**

votes

**0**answers

160 views

### Infinite simple p-groups with only trivial irreps in characteristic p

Is there a prime $p$ and an infinite simple $p$-group $G$ such that for any field $K$ of characteristic $p$ the only irreducible $KG$-module, whether finite or infinite dimensional, is trivial (that ...

**3**

votes

**1**answer

214 views

### For a cross section $\sigma\colon G/N\to G$, how is $\sigma(y)^{-1}\sigma(x)^{-1}\sigma(xy)$ called?

Let $G$ be a locally compact group, let $N$ be a closed normal subgroup of $G$, and let $\sigma\colon G/N\to G$ be a cross section. Let us define $\alpha\colon G/N\times G/N \to N$ by the formula
$$
...

**4**

votes

**1**answer

279 views

### Frequency of a representation of SO(3)

When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In ...

**4**

votes

**1**answer

226 views

### Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power ...

**3**

votes

**0**answers

254 views

### Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...

**1**

vote

**0**answers

131 views

### Does SL(3,q) have a subgroup of order $q^3.(q^3-1)$ [closed]

Let $q=p^n$ for $p>3$.I want to know whether the group $G_2(q)$ has a subgroup of order $q^3.(q^3-1)$. First I look the paper of Peter Kleidman. In this paper Kleidman determined all maximal ...

**2**

votes

**1**answer

169 views

### What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?

**10**

votes

**0**answers

244 views

### Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group
${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use
the low index subgroups ...

**2**

votes

**1**answer

186 views

### About direct limit of groups

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$.
Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$.
Let ...

**3**

votes

**0**answers

161 views

### Alternating quotients of (2,3,7;10)

It was shown that the only quotient of the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ of the form PSL(2, q) is PSL(2, 41). That lead me to consider other families of simple ...

**4**

votes

**1**answer

105 views

### Finiteness properties for graph of groups decompositions

My curiosity was raised by the following question
and the huge variety of comments and suggestions it attracted. I wondered if a converse statement might be equally interesting.
Let $G$ be a finitely ...

**3**

votes

**2**answers

253 views

### If d(“G/H”) < d(G) = 2, must H contain a primitive element?

Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. ...

**18**

votes

**6**answers

1k views

### Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain.
Magnus, in his 1968 ...

**6**

votes

**0**answers

185 views

### Polynomial growth without Gromov's theorem

It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies ...

**1**

vote

**1**answer

72 views

### An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$

In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I ...

**8**

votes

**3**answers

243 views

### How do small central extensions drop the dimension of a faithful representation?

Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon ...

**2**

votes

**1**answer

108 views

### Involution on the components of a group algebra

If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has ...

**10**

votes

**1**answer

488 views

### A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
This question was first posted here.